Constraints, Moment Maps, and Gauge Orbits

A constraint restricts phase space to a subspace, but it does not automatically remove a degree of freedom. Redundancy appears when vector fields generated by constraints are tangent to the constraint surface and lie in the kernel of its pulled-back two-form. Moment maps package this situation for Hamiltonian group actions: their level sets impose constraints, and group orbits supply the candidate gauge directions. The distinction between restriction and quotient is the organizing idea.

Required background. Exterior Algebra and Differential Forms supplies contractions and pullbacks, while the Exterior Derivative and Poincaré Lemma supplies closed two-forms. Tensor Fields and Natural Operations supplies smooth tensors and natural maps; Symmetry Algebras, Orbits, and Fixed Sets supplies the group-action and orbit language.

Helpful background. Calculus and Linear Algebra supplies level sets and kernels; Variational Mechanics and Symmetries supplies constraints and symmetry generators in mechanics.

Constraint surfaces and null directions

Let $(P,\Omega)$ be symplectic and let constraints $C_a:P\to\mathbb R$ define

$$\mathcal C=\{z\in P:C_a(z)=0\}.$$

If zero is a regular value of $C=(C_1,\ldots,C_k)$, then $\mathcal C$ is a smooth codimension-$k$ submanifold. Write $i:\mathcal C\hookrightarrow P$ and $\Omega_\mathcal C=i^*\Omega$. Although $\Omega$ is nondegenerate on $P$, its pullback can be degenerate. A vector $v\in T_z\mathcal C$ belongs to the kernel when $\Omega(v,w)=0$ for every $w\in T_z\mathcal C$.

With $\iota_{X_F}\Omega=dF$, the Hamiltonian vector field $X_{C_a}$ is tangent to $\mathcal C$ precisely when

$$X_{C_a}[C_b]=\{C_b,C_a\}=0$$

on $\mathcal C$. A common way this occurs is that the bracket is a smooth linear combination of the constraints in a neighborhood of $\mathcal C$. For such first-class constraints, $X_{C_a}|_\mathcal C$ annihilates $\Omega_\mathcal C$: any tangent $w$ has $dC_a(w)=0$, hence $\Omega(X_{C_a},w)=0$. The orbit generated by these null directions changes the representative without changing functions constant along the orbit. Second-class constraints instead have an invertible bracket matrix and do not generate kernel directions in this way.

Moment maps for group actions

Suppose a Lie group $G$ acts on $P$ on the left by symplectomorphisms. Fix the fundamental-field convention $\xi_P(z)=\left.d\bigl(\exp(t\xi)\cdot z\bigr)/dt\right|_{t=0}$; with this choice $[\xi_P,\eta_P]=-[\xi,\eta]_P$. A moment map is a map $\mu:P\to\mathfrak g^*$ such that

$$d\langle\mu,\xi\rangle=\iota_{\xi_P}\Omega.$$

Thus $\xi_P=X_{\mu^\xi}$ with $\mu^\xi=\langle\mu,\xi\rangle$. Equivariance means $\langle\mu(gz),\xi\rangle=\langle\mu(z),\operatorname{Ad}_{g^{-1}}\xi\rangle$. Differentiating this equality and using $X_F[G]=-\{F,G\}$ shows that Poisson brackets reproduce the Lie algebra:

$$\{\mu^\xi,\mu^\eta\}=\mu^{[\xi,\eta]}.$$

The zero level $\mu^{-1}(0)$ is invariant, and its characteristic directions are generated by the group action under the regularity hypotheses. Cannas da Silva develops moment maps and their relation to Hamiltonian actions in Cannas da Silva 2006, Lecture 22, pp. 133–140.

Not every symplectic action has a global moment map. The closed one-form $\iota_{\xi_P}\Omega$ may fail to be exact, and equivariance can have an additional obstruction. A local generator should therefore not be silently promoted to a global constraint function.

Rotation on the cotangent plane

Take $P=T^*\mathbb R^2$ with coordinates $(x,y,p_x,p_y)$ and

$$\Omega=dx\wedge dp_x+dy\wedge dp_y.$$

The cotangent lift of a counterclockwise rotation has infinitesimal generator

$$\xi_P=-y\partial_x+x\partial_y-p_y\partial_{p_x}+p_x\partial_{p_y}.$$

For $SO(2)$, whose Lie algebra is one-dimensional, define

$$J=xp_y-yp_x.$$

The canonical formula for $X_J$ gives exactly $X_J=\xi_P$, so $J$ is the moment-map component. This independently checks the sign convention. The level $J=0$ consists of covectors with zero angular momentum. Away from points with enhanced stabilizer, quotienting by rotations removes the angular position together with its null direction on the constraint surface.

At the origin with zero momentum, the whole rotation group fixes the point. The action is not free there, and the quotient has a singular stratum. This elementary example shows why “one first-class constraint removes two phase-space dimensions” is only a regular-region count, not a universal theorem.

Gauge language and its limits

Calling an orbit gauge requires more than a first-class algebra. The transformations must preserve the admissible fields, boundary conditions, and action. In field theory, parameters that vanish suitably at the boundary often generate degeneracies, whereas transformations with nonvanishing boundary action may carry observable data. Boundary conditions decide the distinction; the word gauge does not.

Likewise, irregular constraints can change rank, and nonproper actions can produce non-Hausdorff quotients. A precise version of singular reduction assumes a connected symplectic manifold, a compact Lie group, and an equivariant moment map. At zero momentum, the connected components of the fixed-orbit-type quotients $(P_{(H)}\cap\mu^{-1}(0))/G$ carry symplectic forms, where $P_{(H)}$ consists of points whose stabilizer is conjugate to $H$. These components form the strata of a stratified space rather than necessarily one smooth manifold Sjamaar and Lerman 1991, Introduction, Theorem 2.1, and §6, pp. 2, 9, and 36–44 of the preprint, PDF. Compactness makes the action proper, but properness of the moment map is a different condition. The latter is added for the paper’s stronger connected-open-dense-stratum theorem, not for the orbit-type decomposition just stated.

Exercise: translations as a moment map

On $T^*\mathbb R$ with $\Omega=dq\wedge dp$, let $\mathbb R$ act by $q\mapsto q+a$. Find its moment map and describe the zero-level quotient.

Solution

The generator is $\partial_q$. Since $\iota_{\partial_q}(dq\wedge dp)=dp$, the moment map is $\mu=p$ up to an additive constant. The zero level is the line $p=0$. Pulling back $\Omega$ to that one-dimensional line gives zero, and translations move along its entire length. The quotient is a single point. The action is free and proper here, so the reduced space is smooth and its dimension is $2-2=0$.

References

• Cannas da Silva, Ana. Lectures on Symplectic Geometry. Lecture Notes in Mathematics 1764, revised edition, Springer, 2006. Lectures 22–24, pp. 133–154.
• Sjamaar, Reyer, and Eugene Lerman. “Stratified Symplectic Spaces and Reduction.” Annals of Mathematics 134, no. 2 (1991): 375–422. DOI; Open PDF, November 1990 preprint.